Monday, June 14, 3pm ET (8pm BST, 7am Tue NZST)
Youngho Yoo, Georgia Tech
Packing A-paths of length 0 modulo a prime

Abstract:

An $A$-path is a nontrivial path with its endpoints in a vertex set $A$ that is internally disjoint from $A$. In 1961, Gallai showed that $A$-paths satisfy an approximate packing-covering duality, also known as the Erdős-Pósa property. There are many generalizations and variants of this result. In this talk, we show that the Erdős-Pósa property holds for $A$-paths of length 0 mod $p$ for every prime $p$, answering a question of Bruhn and Ulmer. The proof uses structure theorems for undirected group-labelled graphs. We also give a characterization of abelian groups $\Gamma$ and elements $\ell \in \Gamma$ for which the Erdős-Pósa property holds for $A$-paths of weight $\ell$ in undirected $\Gamma$-labelled graphs. Joint work with Robin Thomas.

Monday, June 7, 3pm ET (8pm BST, 7am Tue NZST)
Abhinav Shantanam, Simon Fraser University
Pancyclicity in $4$-connected planar graphs

Abstract:
A graph on $n$ vertices is said to be pancyclic if, for each $k \in \{3,…,n\}$, it contains a cycle of length $k$. Following Bondy’s meta-conjecture that almost any nontrivial condition on a graph which implies Hamiltonicity also implies pancyclicity, Malkevitch conjectured that a $4$-regular, $4$-connected planar graph containing a $4$-cycle is pancyclic. We show that, for any edge $e$ in a $4$-connected planar graph $G$, there exist $\lambda(n-2)$ cycles of pairwise distinct lengths containing $e$, where $5/12 \leq \lambda \leq 2/3$. Joint work with Bojan Mohar.

Monday, May 31, 3pm ET (8pm BST, 7am Tue NZST)
Daniel Slilaty, Wright State University
Orientations of golden-mean matroids

Abstract:

Tutte proved that a binary matroid is representable over some field of characteristic other than 2 if and only if the matroid is regular. His result inspired the discovery of analogs for $GF(3)$-representable matroids by Whittle, $GF(4)$-representable matroids by Vertigan as well as Pendavingh and Van Zwam, and $GF(5)$-representable matroids by Pendavingh and Van Zwam.

Bland and Las Vergnas proved that a binary matroid’s orientations correspond to its regular representations. (Minty’s result on digraphoids is closely related.) Lee and Scobee proved that a $GF(3)$-representable matroid’s orientations correspond to its dyadic representations. In this talk we will explore orientations of $GF(4)$-representable matroids. A natural partial field to use is the golden-mean partial field; however, not every orientation of a $GF(4)$-representable matroid comes from a golden-mean representation. For example, $U_{3,6}$ has 372 orientations but only 12 of them come from golden-mean representations. We will give a combinatorial characterization of those orientations of $GF(4)$-representable matroids which do come from golden-mean representations and show that these orientations correspond one-to-one to the golden-mean representations.

Joint work with Jakayla Robbins.

Monday, May 17, 3pm ET (8pm BST, 7am Tue NZST)
Bertrand Guenin, University of Waterloo
Graphs with the same even cycles

Abstract: